Semicomplete compositions of digraphs

Abstract

Let T be a digraph with vertices u1,…,ut (t≥2), and let H1,…,Ht be digraphs such that Hi has vertices ui,ji, where 1≤ji≤ni. The composition Q=T[H1,…,Ht] is a digraph with vertex setV(Q)={ui,ji:1≤i≤t,1≤ji≤ni} and arc set(⋃i=1tA(Hi))⋃(⋃uiup∈A(T){ui,jiup,qp:1≤ji≤ni,1≤qp≤np}). The composition Q=T[H1,…,Ht] is a semicomplete composition if T is semicomplete, i.e., there is at least one arc between every pair of vertices. Digraph compositions generalize some families of digraphs, including quasi-transitive digraphs, extended semicomplete digraphs and lexicographic product digraphs. In particular, strong semicomplete compositions form a significant generalization of strong quasi-transitive digraphs.In this paper, we study the structural properties of semicomplete compositions and obtain results on connectivity, paths, cycles and strong spanning subdigraphs. Our results show that this class of digraphs shares some nice properties of quasi-transitive digraphs.